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% Created on 2008-03-27 by ZHENG Zhong
% Last changed at $Date: 2008-04-28 10:15:35 +0000 (Mon, 28 Apr 2008) $ by $Author: heavyzheng $, $Revision: 39 $
% $HeadURL: http://buggarden.googlecode.com/svn/quant/study_notes/martingales_and_measures.tex $
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\chapter{Martingales and Measures}

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\section{The Market Price of Risk}

We start by considering the properties of derivatives dependent on the value of a single variable,
$\theta$. We will assume that the process followed by $\theta$ is:

\begin{equation}
  \frac{d \theta}{\theta} = m dt + s dz
  \label{20080327_process_theta}
\end{equation}

where $dz$ is a Wiener process. The parameters $m$ and $s$ are the expected growth rate in $\theta$
and the volatility of $\theta$, respectively. We assume that they depend only on $\theta$ and time
$t$.

Suppose that $f_1$ and $f_2$ are the prices of two derivatives dependent only on $\theta$ and $t$.
These can be options or other instruments that provide a payoff equal to some function of $\theta$
at some future time. We assume that during the time period under consideration, $f_1$ and $f_2$
provide no income.

Suppose that the processes followed by $f_1$ and $f_2$ are:

\[ \frac{df_1}{f_1} = \mu_1 dt + \sigma_1 dz \]
\[ \frac{df_2}{f_2} = \mu_2 dt + \sigma_2 dz \]

where $\mu_1$, $\mu_2$, $\sigma_1$ and $\sigma_2$ are functions of $\theta$ and $t$. The $dz$ is the
same Wiener process as in equation \eqref{20080327_process_theta}, because it is the only source of
the uncertainty in the prices of $f_1$ and $f_2$.

\textcolor{red}{$f_1$ and $f_2$ are functions of $\theta$ and $t$. How can we get the processes of
$f_1$ and $f_2$ as above? Using It\^o's lemma or deriving from the preceding chapter?}

The discrete versions of the processes are:

\begin{equation}
  \Delta f_1 = \mu_1 f_1 \Delta t + \sigma_1 f_1 \Delta z
  \label{20080327_Delta_f_1}
\end{equation}

\begin{equation}
  \Delta f_2 = \mu_2 f_2 \Delta t + \sigma_2 f_2 \Delta z
  \label{20080327_Delta_f_2}
\end{equation}

We can eliminate the $\Delta z$ by forming an instantaneously riskless portfolio consisting of
$\sigma_2 f_2$ of the first derivative and $-\sigma_1 f_1$ of the second derivative. If $\Pi$ is the
value of the portfolio, then:

\begin{equation}
  \Pi = (\sigma_2 f_2) f_1 - (\sigma_1 f_1) f_2
  \label{20080327_Pi}
\end{equation}

and:

\[ \Delta \Pi = \sigma_2 f_2 \Delta f_1 - \sigma_1 f_1 \Delta f_2 \]

Substituting from equations \eqref{20080327_Delta_f_1} and \eqref{20080327_Delta_f_2}, we obtain:

\begin{equation}
  \Delta \Pi = (\mu_1 \sigma_2 f_1 f_2 - \mu_2 \sigma_1 f_1 f_2) \Delta t
  \label{20080327_Delta_Pi}
\end{equation}

Because the portfolio is instantaneously riskless, it must earn the risk-free rate. Hence:

\[ \Delta \Pi = r \Pi \Delta t \]

Substituting into this equation from equations \eqref{20080327_Pi} and \eqref{20080327_Delta_Pi}
gives:

\[ \mu_1 \sigma_2 - \mu_2 \sigma_1 = r \sigma_2 - r \sigma_1 \]

or:

\[ \frac{\mu_1 - r}{\sigma_1} = \frac{\mu_2 - r}{\sigma_2} \]

Define $\lambda$ as the value of each side in the above equation, so that:

\[ \frac{\mu_1 - r}{\sigma_1} = \frac{\mu_2 - r}{\sigma_2} = \lambda \]

Dropping subscripts, we have shown that, if $f$ is the price of a derivative dependent only on
$\theta$ and $t$ with:

\begin{equation}
  \frac{df}{f} = \mu dt + \sigma dz
  \label{20080327_process_f}
\end{equation}

then:

\begin{equation}
  \frac{\mu - r}{\sigma} = \lambda
  \label{20080327_lambda_of_theta}
\end{equation}

The parameter $\lambda$ is known as the \emph{market price of risk} of $\theta$. Note that $\lambda$
can be dependent on both $\theta$ and $t$, but it is not dependent on the nature of the derivative
$f$. At any given time, $(\mu - r) / \sigma$ must be the same for all derivatives that are dependent
only on $\theta$ and $t$.

\textcolor{red}{todo: see p485.}

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\section{Traditional Risk-Neutral World and Alternative Worlds}

The risk-neutral valuation principle states that a derivative can be valued by (1) calculating the
expected payoff on the assumption that the expected return from the underlying asset equals the
risk-less interest rate, ant (2) discounting the expected payoff at the risk-free interest rate.

From equation \eqref{20080327_process_f}, the process followed by derivative price $f$ is:

\[ df = \mu f dt + \sigma f dz \]

The value of $\mu$ depends on the risk preferences of investors. In a world where the market price
of risk is zero, $\lambda$ equals zero. From equation \eqref{20080327_lambda_of_theta}, $\mu = r$,
so that the process followed by $f$ is:

\[ df = r f dt + \sigma f dz \]

This is referred to as the \emph{traditional risk-neutral world}.

By making other assumptions about the market price of risk, we define other worlds that are
internally consistent. In general, when the market price of risk is $\lambda$, equation
\eqref{20080327_lambda_of_theta} shows that:

\[ \mu = r + \lambda \sigma \]

so that:

\[ df = (r + \lambda \sigma) f dt + \sigma f dz \]

The market price of risk of a variable determines the growth rates of all securities dependent on
the variable. As we move from one market price of risk to another, the expected growth rates of
security prices change, but their volatilities remain the same. \textcolor{red}{illustrated in
section 10.7.} Choosing a particular market price of risk is also referred to as defining the
\emph{probability measure}. For some value of the market price of risk, we obtain the ``real world''
and the growth rates of security prices that are observed in practice.

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\section{Generalizations of It\^o's Lemma}

It\^o's Lemma provides the process followed by a function of a single stochastic variable. Here we
present a generalized version of It\^o's lemma for the process followed by a function of several
stochastic variables.

Suppose that a function $f$ depends on the $n$ variables $x_1, x_2, ..., x_n$ and time $t$:

\[ f = f(x_1, x_2, ..., x_n, t) \]

Suppose further that $x_i$ follows an It\^o process with instantaneous drift rate $a_i$ and
instantaneous variance $b_i^2$ ($1 \leq i \leq n$), that is:

\begin{equation}
  dx_i = a_i dt + b_i dz_i
  \label{20080327_dx_i_continuous}
\end{equation}

where $dz_i$ ($1 \leq i \leq n$) is a Wiener process. Each $a_i$ and $b_i$ may be any function of
all the $x_i$'s and $t$.

A Taylor series expansion of $f$ gives:

\begin{equation}
  \Delta f = \sum_{i=1}^n \frac{\partial f}{\partial x_i} \Delta x_i
           + \frac{\partial f}{\partial t} \Delta t \\
           + \frac{1}{2}
             \sum_{i=1}^n \sum_{j=1}^n
             \frac{\partial^2 f}{\partial x_i \partial x_j} \Delta x_i \Delta x_j
           + \frac{1}{2}
             \sum_{i=1}^n
             \frac{\partial^2 f}{\partial x_i \partial t} \Delta x_i \Delta t \\
           + \cdots
  \label{20080327_Delta_f_taylor_exp}
\end{equation}

Equation \eqref{20080327_dx_i_continuous} can be discretized as:

\[ \Delta x_i = a_i \Delta t + b_i \epsilon_i \sqrt{\Delta t} \]

where $\epsilon_i$ is a random sample from a standardized normal distribution. The correlation
$\rho_{ij}$ between $dz_i$ and $dz_j$ is defined as the correlation between $\epsilon_i$ and
$\epsilon_j$. \textcolor{red}{see appendix 11A} It was argued that:

\[ \lim_{\Delta t \to 0} \Delta x_i^2 = b_i^2 dt \]

Similarly:

\[ \lim_{\Delta t \to 0} \Delta x_i \Delta x_j = b_i b_j \rho_{ij} dt \]

As $\Delta t \to 0$, the first three terms in the expansion of $\Delta f$ in equation
\eqref{20080327_Delta_f_taylor_exp} are of order $\Delta t$. All other terms are of higher order.
Hence:

\[
  df = \sum_{i=1}^n \frac{\partial f}{\partial x_i} dx_i
     + \frac{\partial f}{\partial t} dt
     + \frac{1}{2}
       \sum_{i=1}^n \sum_{j=1}^n
       \frac{\partial^2 f}{\partial x_i \partial x_j} b_i b_j \rho_{ij} dt
\]

This is the generalized version of It\^o's lemma. Substituting for $dx_i$ from equation
\eqref{20080327_dx_i_continuous} gives:

\begin{equation}
  df = \Big(
           \sum_{i=1}^n \frac{\partial f}{\partial x_i} a_i
         + \frac{\partial f}{\partial t}
         + \frac{1}{2}
           \sum_{i=1}^n \sum_{j=1}^n
           \frac{\partial^2 f}{\partial x_i \partial x_j} b_i b_j \rho_{ij}
       \Big) dt
     + \sum_{i=1}^n \frac{\partial f}{\partial x_i} b_i dz_i
\end{equation}

\textcolor{red}{An alternative generalization: see p504.}

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\section{Several State Variables}

\textcolor{red}{todo: see p487.}

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\section{Martingales}

A martingale is a zero-drift stochastic process. A variable $\theta$ follows a martingale if its
process has the form:

\[ d\theta = \sigma dz \]

where $dz$ is a Wiener process. The variable $\sigma$ may itself be stochastic. It can depend on
$\theta$ and other stochastic variables.

A martingale has the convenient property that its expected value at any future time is equal to its
value today. This means:

\[ E(\theta_T) = \theta_0 \]

where $\theta_0$ and $\theta_T$ denote the values of $\theta$ at times zero and $T$, respectively.

\textcolor{red}{todo: see p488.}

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